A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

1, 2, 3, 4, 6, 7, 5, 9, 13, 14, 8, 12, 17, 24, 25, 10, 16, 23, 29, 40, 41, 11, 19, 28, 39, 46, 62, 63, 15, 22, 32, 45, 61, 69, 91, 92, 18, 27, 38, 50, 68, 90, 99, 128, 129, 20, 31, 44, 60, 74, 98, 127, 137, 174, 175, 21, 34, 49, 67, 89, 105, 136, 173, 184, 230, 231, 26, 37, 53, 73, 97, 126, 144, 183, 229, 241, 297, 298

Offset

1,2

Comments

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186004 lists those n for which j=n, the distance from (i,j,k) to the xz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186004 is a permutation of the positive integers.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Example

Northwest corner:
   1,  2,  4,  5,  8, 10
   3,  6,  9, 12, 16, 19
   7, 13, 17, 23, 28, 32
  14, 24, 29, 39, 45, 50
  25, 40, 46, 61, 68, 74
T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...

Mathematica

lexicographicLattice[{dim_, maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1, {dim}], 1]&, maxHeight], 1];
lexicographicLatticeHeightArray[{dim_, maxHeight_, axis_}]:=Array[Flatten@Position[Map[#[[axis]]&, lexicographicLattice[{dim, maxHeight}]], #]&, maxHeight];
llha=lexicographicLatticeHeightArray[{3, 12, 2}];
ordering=lexicographicLattice[{2, Length[llha]}];
llha[[#1, #2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

Crossrefs

Cf. A057557, A186003, A186005.

Sequence in context: A026236 A238862 A048201 * A056534 A297933 A360646

Adjacent sequences: A186001 A186002 A186003 * A186005 A186006 A186007

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 10 2011

Status

approved